Uniformization of compact complex manifolds by Anosov homomorphisms
Uniformization of compact complex manifolds by Anosov homomorphisms
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DOI：
10.1007/s00039-021-00572-6
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发表时间：
2021-08
影响因子：
2.2
通讯作者：
David Dumas;Andrew Sanders
中科院分区：
文献类型：
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作者：
David Dumas;Andrew Sanders
我们研究了紧致流形的一致化问题，这些流形是由Anosov同态的图像作为复旗簇中域的继承者而产生的。我们专注于Anosov同态与“小”的极限集，作为衡量的黎曼豪斯多夫余维的标志品种。在这样的余维假设下，我们证明了在相关的紧致复流形上的复结构的所有一阶变形都是通过Anosov同态的变形实现的。通过一些温和的额外假设，我们证明了特征簇局部同胚映射到流形的（广义）Teichmüller空间。特别是，这在Anosov同态到更高秩复半单李群的背景下提供了Bers同时一致化定理的局部模拟。
We study uniformization problems for compact manifolds that arise as quotients of domains in complex flag varieties by images of Anosov homomorphisms. We focus on Anosov homomorphisms with “small” limit sets, as measured by the Riemannian Hausdorff codimension in the flag variety. Under such a codimension hypothesis, we show that all first-order deformations of complex structure on the associated compact complex manifolds are realized by deformations of the Anosov homomorphism. With some mild additional hypotheses we show that the character variety maps locally homeomorphically to the (generalized) Teichmüller space of the manifold. In particular this provides a local analogue of the Bers Simultaneous Uniformization Theorem in the setting of Anosov homomorphisms to higher-rank complex semisimple Lie groups.